Fraction Comparison Challenge Activity

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I was sitting in a fifth grade math class yesterday afternoon watching a teacher do what I thought was pretty good lesson on fractions. As the kids struggled to figure out how to compare fractions, I was asked by a student whether fractions that had small denominators were larger. I answered, “well, that really depends....” and without missing a beat, that same student said, “so, if the numerator is big, then the fractions must be big too....” I replied, “well, that depends as well....” to which the student threw up her arms and said, “Robert, I can’t take this anymore!”

Well, neither can I....

So I came up with the activity you see attached: it’s meant to help students understand that the size of a fraction depends on the relationship between the numerator and denominator, not just the sizes. So yes, halves are bigger than thirds, but that doesn’t mean every fraction with halves if more than fractions with thirds: look at 1/2 vs. 2/3, or 0/2 vs. 1/3.

Thus, an activity was born: why not give the students a “piece” of the fraction, tell them to make it “big” or “small” and then have them put in a number that would make it big or small. That is, why not turn writing fractions into a problem solving activity? Mais, oui, monsieur!

Look at the example on the left: how can we create two fractions that have the same denominator, and a somewhat small number at that, and turn it into both a big AND small fraction (and, to make it more annoying, let’s omit 0 and improper fractions....) In this cae, 1/3 is the small fraction and 3/3 would be the big fraction.

The follow up question would be this: how can we make a fraction that is even smaller than 1/3, without using 0 or less (like a negative fraction)?

If you have wise-guy students (and we all do, don’t we?), then some will say, “well, let’s put a really small number in the numerator, and it will get smaller and smaller, won’t it?” So, yes, we can put in 1/3 in the numerator, and that indeed would be 1/3 thirds, which is even smaller (and which goes by the name “complex fraction,” btw.) All of which means they understand that it’s just not the thirds that count, but also the number of them.....

Things can get more interesting if we start playing with the numbers in the numerator and denominator. Take a look at the next puzzle:

Now, kids know that 1 of something is going to be bigger than 5 of something, but what they (hopefully) will see here is that depends on what those “somethings” are. 1/3 is going to be much bigger than 5/30, because even though there are 5 times more pieces selected in the second number, 30ths are much, much smaller than 3rds.

Similarly, in the example on the right, your students (should) know that 9ths are smaller than 5ths. So if 5ths are bigger pieces, how can I make sure that the fraction is smaller than one that has 9ths? Well, I would have to make it a small number of 5ths, and a large number of 9ths.....

Anyway, that’s how this activity goes; I think it’s pretty cool and I hope you’ll use it in all sorts of neat ways and then tell me how it turned out, okay?

On the last two pages of this document, you’ll see that there are fields which you can highlight and put in your own set of numbers. Please note that this is optimized for Adobe Acrobat Reader, so if you’re using a different PDF viewer, the layout of the numbers may get messed up. Hey, Adobe Acrobat Reader is free, and you’re probably using it already, right? If not, there are also blank versions of this activity where you can hand write the numbers.

One really cool thing you can do is to make this into a worksheet by adding all the numbers you want, and then printing 4 on a page, which can easily be done using the print options dialogue. Don’t know how to do that? That’s why they have Google!